3|n - 4| = 15What is the positive solution to the given equation?

1. TRANSLATE the problem information

• Given: \(3|\mathrm{n} - 4| = 15\)
• Find: The positive solution for n

2. INFER the solution approach

• We need to isolate the absolute value expression first
• Then apply the fundamental property: \(|\mathrm{A}| = \mathrm{B}\) means \(\mathrm{A} = \mathrm{B}\) or \(\mathrm{A} = -\mathrm{B}\)

3. SIMPLIFY to isolate the absolute value

• Divide both sides by 3:

\(3|\mathrm{n} - 4| = 15\)

\(|\mathrm{n} - 4| = 5\)

4. CONSIDER ALL CASES for the absolute value equation

• Since \(|\mathrm{n} - 4| = 5\), we have two possibilities:

Case 1: \(\mathrm{n} - 4 = 5\)

• Add 4 to both sides: \(\mathrm{n} = 9\)

Case 2: \(\mathrm{n} - 4 = -5\)

• Add 4 to both sides: \(\mathrm{n} = -1\)

5. APPLY CONSTRAINTS to select final answer

• We found two solutions: \(\mathrm{n} = 9\) and \(\mathrm{n} = -1\)
• The question asks specifically for the positive solution
• Therefore: \(\mathrm{n} = 9\)

Answer: D) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Forgetting that absolute value equations have two cases

Students might isolate \(|\mathrm{n} - 4| = 5\) correctly, but then only solve \(\mathrm{n} - 4 = 5\), getting \(\mathrm{n} = 9\). While this happens to give the correct answer, they miss the complete solution process. More problematically, some students might only solve \(\mathrm{n} - 4 = -5\), getting \(\mathrm{n} = -1\), then think there's no positive answer and resort to guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Skipping the isolation step

Students might try to solve \(3|\mathrm{n} - 4| = 15\) by thinking \(|\mathrm{n} - 4| = 15\), leading to \(\mathrm{n} - 4 = \pm 15\). This gives \(\mathrm{n} = 19\) or \(\mathrm{n} = -11\), and they would select Choice E (19) as the positive solution.

The Bottom Line:

This problem tests whether students understand that absolute value equations systematically produce two cases, and whether they can maintain organization through multi-step algebraic manipulation while applying constraints to select the appropriate final answer.